Blender renders always frame 1 on 2nd computer
I answer the question on myself. I can't excatly describe what the problem was, but I...

a) updated from Blender 2.79 to 2.80 beta
b) baked animations and physics
c) corrected dope sheet animations with start and end points

As side effect I have a ~5x performace gain, what is great, I'm not sure if the new blender or the baked animations did this.
Now parallel computation on 2 notebooks and 1 desktop runs fine, resulting in another ~3x performance gain. :-)